7.15.24 problem 24

Internal problem ID [480]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 24
Date solved : Tuesday, March 04, 2025 at 11:25:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+2 x y^{\prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

Order:=6; 
ode:=3*x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {1}{14} x^{2}+\frac {1}{728} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {1}{10} x^{2}+\frac {1}{440} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 47

ode=3*x^2*D[y[x],{x,2}]+2*x*D[y[x],x]+x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {x^4}{728}-\frac {x^2}{14}+1\right )+c_2 \left (\frac {x^4}{440}-\frac {x^2}{10}+1\right ) \]

✓ Sympy. Time used: 0.894 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + 3*x**2*Derivative(y(x), (x, 2)) + 2*x*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{440} - \frac {x^{2}}{10} + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {x^{4}}{728} - \frac {x^{2}}{14} + 1\right ) + O\left (x^{6}\right ) \]